Markov Processes with Jumps on Manifolds and Lie Groups

TL;DR

This paper reviews recent advances in the theory of Markov processes with jumps on manifolds and Lie groups, focusing on stochastic differential equations, the Courrège theorem, and invariant processes under group actions.

Contribution

It synthesizes developments in Markov processes with jumps within geometric settings, highlighting new results on stochastic equations and invariance properties.

Findings

Analysis of stochastic differential equations in Markus form

Extension of Courrège theorem to Lie groups

Characterization of invariant Markov processes on manifolds

Abstract

We review some developments concerning Markov and Feller processes with jumps in geometric settings. These include stochastic differential equations in Markus canonical form, the Courr\`{e}ge theorem on Lie groups, and invariant Markov processes on manifolds under both transitive and more general Lie group actions.